One venue that Republicans are turning to for leverage, starting next week: oversight hearings, beginning with some tough questions about why the rollout of the website for enrolling in health care exchanges is having so many problems.

"The American people deserve to know what caused this mess," said Rep. Fred Upton, R-Michigan, chairman of the House Energy and Commerce Committee. "Delays and technical failures have reached epidemic proportions."

Upton has already asked officials from the Department of Health and Human Services to brief his committee on the problems at a hearing scheduled for next Thursday. He also is asking them to turn over records of their communications with website contractors about the preparations for the site's launch, and the problems people have had trying to use it.

Additionally, the chair of the House Oversight and Government Reform Committee, Rep. Darrell Issa, last week wrote to Health and Human Services Secretary Kathleen Sebelius demanding information about the rollout problems. Joining him was Sen. Lamar Alexander, ranking Republican on the Senate committee on health, education, labor and pensions.

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One issue has been the ability to log in to healthcare.gov. A senior administration official told CNN some users, especially those who signed up "early on," have been having trouble logging in, but the administration is working on a fix. There also was conflicting word on whether some passwords were deleted if they were created in the first week or so after the launch.

Sebelius concedes there have been implementation difficulties.

"I'll be the first to tell you that the website launch was rockier than we would have liked," she told an audience in Cincinnati on Wednesday, during her tour to promote Obamacare. But, she told CNN affiliate WLWT, "There are constant improvements under way, so that we are getting people in much more quickly."

Nevertheless, a couple of Republicans have called on Sebelius to resign.

"Enough is enough," said Sen. Pat Roberts, who has called for her to step down in spite of being a longtime friend of her family. "Secretary Sebelius has had three and a half years to launch Obamacare, and she has failed."

Rep. John Fleming of Louisiana tells CNN he is working to collect signatures calling for Sebelius to leave.

White House spokesman Jay Carney pushed back on Tuesday.

"The secretary does have the full confidence of the president," he told reporters. And when he was asked Thursday who would be held accountable for the problems with the website, Carney replied, "The people who are responsible for making it work are hard at work, fixing the problems that need to be fixed."

He focused instead on the benefits the uninsured Americans are now entitled to.

"The result will be millions of Americans who have insurance who did not have it before. That's the goal. And the goal is not about the website. The goal is that the American people who have been shut out of affordable health care options in the past have those options available to them."

But former White House spokesman Robert Gibbs, while not naming names, said of the website on MSNBC, "when they get it fixed, I hope they fire some people that were in charge."

Public attention may have been focused more on the shutdown fight this month than the problems with enrolling in the exchanges. But with the shutdown ended, part of the emerging Republican message is that the problems with the Obamacare website reflect broader problems with the law overall.

"I think this is emblematic of how problematic this is going to be in the future," said Fleming. At a minimum, the implementation of the individual mandate, requiring people to get health care, should be delayed, he said.

But a spokeswoman for Sebelius said the online enrollment process is proceeding.

"While traffic is down somewhat from its peak on day one, it remains high as Americans continue to seek to learn more about their new coverage options," said spokeswoman Joanne Peters.